To link to the entire object, paste this link in email, IM or documentTo embed the entire object, paste this HTML in websiteTo link to this page, paste this link in email, IM or documentTo embed this page, paste this HTML in website

Eigenfunctions for Random Walks on Hyperplane Arrangements
by
John Pike
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Ful llment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(APPLIED MATHEMATICS)
August 2013
Copyright 2013 John Pike

A large class of seemingly disparate Markov chains can be modeled as random walks on the chambers of hyperplane arrangements. Examples include models from computer science, statistical mechanics, military campaigns, and card shuffling, as well as many natural random walks on finite reflection groups. A remarkable feature of these chains is that the general theory is fairly complete. For instance, it has been established that the corresponding transition matrices are diagonalizable over the reals with explicitly computable nonnegative eigenvalues of known multiplicity. Moreover, a description of the stationary distributions and a criterion for ergodicity are known, and there are simple upper bounds on the rates of convergence. ❧ The present work continues these investigations by providing an explicit construction of the right eigenfunctions corresponding to the largest eigenvalues and a general prescription for finding others. For certain important classes of chamber walks, we are able to provide a basis for the eigenspace corresponding to the subdominant eigenvalue and we show that several interesting statistics arise in this fashion. In addition, we demonstrate how these eigenfunctions can be used to obtain upper and lower bounds on the variation distance to stationarity. We also discuss connections with Stein's method, including as an aside a derivation of some of the eigenfunctions for certain random walks on the symmetric group which do not fit into the hyperplane paradigm. Along the way, we give generalizations and alternate proofs of some known results and introduce a novel combinatorial description of hyperplane walks which we have found to be quite useful.

The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the author, as the original true and official version of the work, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright. The original signature page accompanying the original submission of the work to the USC Libraries is retained by the USC Libraries and a copy of it may be obtained by authorized requesters contacting the repository e-mail address given.

Eigenfunctions for Random Walks on Hyperplane Arrangements
by
John Pike
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Ful llment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(APPLIED MATHEMATICS)
August 2013
Copyright 2013 John Pike